ON the CLASSIFICATION of ISOTROPIC TENSORS by P

Home , Cartesian tensor, Kronecker delta

ON THE CLASSIFICATION OF ISOTROPIC TENSORS by P. G. APPLEBY, B. R. DUFFY and R. W. OGDEN (Received 14 January, 1986)

A tensor is said to be isotropic relative to a group of transformations if its components are invariant under the associated group of coordinate transformations. In this paper we review the classification of tensors which are isotropic under the general linear group, the special linear (unimodular) group and the rotational group. These correspond respectively to isotropic absolute tensors [4,8], isotropic relative tensors [4] and isotropic Cartesian tensors [3]. New proofs are given for the representation of isotropic tensors in terms of Kronecker deltas and alternating tensors. And, for isotropic Cartesian tensors, we provide a complete classification, clarifying results described in [3]. In the final section of the paper certain derivatives of isotropic tensor fields are examined.

1. Introduction and notation. Let X be a vector space of finite dimension N s= 2 and let X* be its dual. A tensor T of contravariant order U and covariant order V is an element of the vector space vX = X ® X . . . X® X* X* . . . X*

U times V times of dimension Nu+V. We denote by T'fifc;'^ the components of T with respect to a general basis {e,} for Xand dual basis {e1} for X*. All indices run over values 1,2,... ,Nand in what follows the usual Einstein summation convention is adopted. The general linear group of transformations of X to itself is denoted by GL(N); each transformation a e GL(N) defines an associated linear transformation

such that for all \u\2,. . . ,\veXand all yuy2,. . . ,yv eX*

~~.. .<8>yvb, where b is the inverse of a. We call aT the image of T under a. The components of T and aT relative to the basis {e,} are related by~~

~~U V iy . • iu — FT "j fmim 2 FT h' l • •)v 11 ji, n,n2- ..nv r=l~~

~~where the components a) are defined by ae, = a{e; and b] by e'b = b'p'. A tensor T is said to be isotropic relative to a group ^ of non-singular linear Glasgow Math. J. 29 (1987) 185-196.~~

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 186 P. G. APPLEBY, B. R. DUFFY AND R. W. OGDEN transformations X->X if T is invariant under all transformations aeS, that is T = aT Vae<£ (1.2) Since the components of aT relative to the transformed basis {e,} and its dual {e'}, defined by

~~e; = ae; = ^y, e' = e'b = b'fif, are the same as the components of T relative to the basis {e,}, the components of T are invariant under the basis transformation e, >-> ae, for all ae'S.~~

2. General isotropic tensors. Tensors isotropic under the general linear group GL(N) have components that are invariant under all basis transformations. The basic properties of these tensors have been determined by Thomas [8] and Knebelman [4]. Here we review their results and present a new proof of the representation of general isotropic tensors. First we note that any non-trivial general isotropic tensor must have equal contravariant and covariant orders [8,4]. If T is a tensor of type (U, V) (that is, of contravariant order U and covariant order V) then by setting a = Al, where A e U - {0} and 1 denotes the identity transformation, we obtain from (1.1)

~~Hence, if T is isotropic, we must have~~

for all A =£0; hence U = V. If T is a general isotropic tensor of type (U, U) then its components must satisfy u u i, rpmt. mv FT n,jix...iu n nv..nu r=l r=l p p p for all a e GL(N). Differentiating (2.1) with respect to a q and then setting a q = b q, where 6? is the Kronecker delta, we obtain the result

UP1 Jili-Ju^ UP* JVi-Ju^ ••• T UP 1 )\J2-W uJi* Pli-lu T "n1 J\P lu T ••• T ulu1 IM2-P given by Thomas [8]. Thomas outlined a method in which a system of linear equations obtained from (2.2) were solved to show, by induction, that T can be written as the sum of products of U Kronecker deltas. A much shorter proof of Thomas's results can be obtained using the following generalization of (2.2). We consider first the case U = 2, for which equation (2.1) becomes / irrkl ^fn^n'rii

~~r Differentiation of this with respect to a s and a" in turn leads to~~

uruu1 pq ~ uu'Jr1 pq upuq1 ru ' upuq1 ur- \^-J) Contraction of u with v in (2.3) yields equation (2.2) for U = 2. We now generalize this formula by differentiating (2.1) with respect to each of Oq\, 0q\, • • • , Oq" in turn; after some rearrangement, we obtain

~~r V L\hh-iu 1 ro(fllq^...qu) V \a(qiqi...q u)'r'\il-'UL OA\ £j *oQ>,p2...pu) )\h-)u Zy ^iih-iu o(pip2-.pu)' K*"^) oeSu oeSu where~~

ahj2-ju~ °h°h • • • °w y1--*) and the summations are over the set Sy of all U\ permutations a of U indices, a iP\Pi • • • Pv) representing a permutation oi(px, p2, . . . , pu). Further identities can be obtained from (2.4) by contraction of 1 or 2 or up to (U — 1) indices pr with qr. The latter of these alternatives recovers (2.2). We now use (2.4) to obtain explicit forms for general isotropic tensors. (i) For the case U = 1 each of (2.2) and (2.4) reduces to ^77 = 5/7% Setting i=p = 1 and relabelling the indices, we obtain the well-known result T^kd), (2.6) x k where A is an arbitrary scalar such that A = T\ = T\ = . . . = T% = N~ T k. (ii) For U = 2, we have from equation (2.3) i (2.7)

~~after setting / = r = 1 and j = u = 2 and relabelling, where A and fi are arbitrary scalars such that~~

Equation (2.7) gives the most general form for the components of an isotropic tensor of type (2,2). (iii) For the general case with if ^ N we set ir =pr = r {r = 1, 2,. . ., U) in (2.4) and relabel suffices to obtain the result

~~1 A hh- Ju~ ZJ o&ju2...jv , K^-o) aeSu~~

~~where the ACT are U\ separate scalars given by j _ rr\2...U cy n\~~

(iv) The result (2.8) also holds when U>N, although the scalars ACT are not then given by (2.9). It suffices to consider the case U = N +1. If U = N + 1 then amongst the indices qx, q2,. • • , qN+l at least one value in the set {1,2, . . . , N} must be repeated. Without loss of generality suppose qN = qN+i. Then, on selecting ir = pr = r(l =£ r =£ N), is+\ = PN+I = N we obtain from (2.4)

~~•Tqy..qNqN+\ _i_ fqi-qN+\iN — V \o(ql...qN+i)'pl2...NN aeSu~~

~~Since qN = qN+x it follows that (2.8) holds with~~

1 _ 1T12. ..AW Aa — 21 o(12...NN)- In a similar way (2.8) follows tor U^N + 2. Note, however, that not all the components A}J^;/.)v are independent when U> N, since ) 2 (sgna)Al^V' = 0 (r&N + l). (2.10) creS, This can be seen by noting that at least one index in the set {1,2, . .., N} must be repeated in iu i2, . .., ir; interchange of an index with its repetition changes the sign of a but does not affect the value of A^'2);'^. Alternatively, the result (2.10) can be obtained by making appropriate choices of T'j1'^;'^ as products of Kronecker deltas in (2.4) and contracting. It is worth noting that it follows from (2.10) that

~~for the components of any tensor T of type (r, r).~~

3. Isotropy under the unimodular group. The unimodular (or special linear) group SL(N) is the subgroup of transformations a e GL(N) such that deta = l. (3.1) A tensor T of type ((/, V) that is isotropic under this group must satisfy the invariance requirement (1.2) for all a e SL(N). Let GL(N)+ be the subgroup of GL(N) consisting of transformations with positive determinant. Then, for each a e GL(N)+ there is an associated unimodular transformation a defined by a = (deta)-lwa. (3.2) From (1.1) we deduce that

~~and hence aT = T for all~~

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 ISOTROPIC TENSORS 189 if and only if (deta)"H'aT = T for all aeGL(N)+, (3.3) where W = (U-V)/N. (3.4) In index form (3.3) can be rearranged as 3 5 rj;::#= (deta)" ft <" ft W;::X- ( - ) r=l s = l The set of tensors T that are isotropic with respect to SL(N) therefore characterizes the set of relative tensors^ that are isotropic with respect to GL(N)+. We shall say that a tensor of type (U, V) that is isotropic with respect to SL(N) has weight W, given by (3.4). It was shown in [4] that for a relative tensor to be isotropic (with respect to GL(N)+) W must be an integer. We now provide a different proof of this result which yields some additional information. The case N = 3 has been discussed in [5]. Let a'j = Xjd'j (no summation), where 0< A, <°° for each i e {1, 2, . . ., N}. Equation (3.5) then reduces to N 'TM-'O 't-'u— FT i W-U, + V,'pi1 v..iu hi~ 11 A l hih-iv'

where Ur (respectively Vr) is the number of times the index r appears in the set {i\, h, • • • , iu} (respectively {jx, j2, . . . , jv}), so that [/ = (/, + U2 +. . . + UN, V = V, + V2 + . . . + VN. Since the kr can be chosen independently, we deduce that

W = Ur-Vr for each r e {1, 2, . . . , N}, (3.6) and hence W is an integer (positive, negative or zero). Equation (3.4) is recovered on summing equations (3.6) for r = l to N. Equations (3.6) are apparently new. We note, in particular, that the only non-zero components of T are those for which Ur — Vr = W for r = 1,2, . . . , N. For example, if T has equal covariant and contravariant order, so that W = 0, then Ur = Vr(r = 1,2, ..., N); thus on a non-zero component 7^;;:^ the covariant indices are a permutation of the contravariant ones, a result that is also evident from (2.8). In this case T is isotropic with respect to GL(N)+. If T is purely contravariant (V = 0) or covariant (U = 0) the order of T is N \W\ and on a non-zero component of T the indices must include each of the integers 1,2,... ,N exactly \W\ times; the particular case |W\ = 1 has special significance, as we see in what follows.

fThe terminology density tensor is also used. See, for example, [2] and [7]. A relative tensor of weight W with components T'^"W with respect to the basis {e,} has components with respect to {§,} given by the right-hand side of (3.5). We shall say that such a relative tensor is of type (U, V, W).

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 190 P. G. APPLEBY, B. R. DUFFY AND R. W. OGDEN In considering isotropy under the unimodular group we may suppose, without loss of generality, that X is an oriented vector space, the orientation being characterized by a particular TV-covector (or volume element) ee f\N X* (the space of N-covectors on X). This structure determines (see, for example, [1]) a special class of bases with respect to each of which the components of e are given by

~~f +1 if i1i2. • . J/v is an even permutation of (1, 2, . . . , N)~~

e,1,2...,JV =-s — 1 if/i/2 • • • ijv is an odd permutation of (1,2,. . . ,N) (3.7) 1^0 otherwise. The dual of e is an iV-vector e* e A^^ witn components, denoted by e'1'2'"'", having the same numerical values as e,,,2 ,N. Since, for each a e GL(N), ae = (deta)-1e, ae* = (deta)e*, l " ' it follows that e and e* are isotropic tensors of order N with respect to SL(N), of weights -1 and +1 respectively. From (3.3) or (3.5) we deduce that the tensor product of two tensors that are isotropic with respect to SL(N) and of weights W and W is isotropic of weight W + W, and any contraction of a pair of co- and contravariant indices of an isotropic tensor yields an isotropic tensor of the same weight. We now prove the following:

THEOREM (cf. Knebelman [4]). Any tensor T that is isotropic with respect to SL(N) can be represented as the product of \W\ of the tensors e (or e*) and the general isotropic tensor with components given by (2.8). (i) If U > V, and hence W > 0, the tensor

~~W times is isotropic of weight zero and type (U, U), and can therefore be represented in the form (2.8). On use of the result~~

~~which follows from the definition (3.7), we conclude that~~

~~fh-iu— V 2 \a('\-iu)piv-n-)v+N JU-N-H-JU (-1 Q\ 1 h-iv~ ZJ A-o^h-iu ^ ...e , jj.y; aeSu w a factor (N\)~ having been absorbed by ACT. (ii) If U < V similar arguments to those used in (i) lead to~~

~~aeSv~~

'it—'u 4- 4- A't/T'i'V-? _ fti T'l- ••'£/ _ &Euclidean vector space, that is X has a positive definite scalar product. In this case we may identify X canonically with its dual space X*, and no distinction need be made between covariant and contravariant tensors. A tensor of order U is simply an element of the vector space

U times Provided we restrict attention to orthonormal basis vectors it is also unnecessary to distinguish between covariant and contravariant components. With this restriction a tensor T e <8>u X is commonly referred to as a Cartesian tensor of order U. The (Cartesian) components of T are denoted by ThJu. The invariance requirement (1.2) for an isotropic Cartesian tensor of order U can be written [3]

~~TV.,,, = aidlahh . . . aiujuTh,,,h, (4.1) where aikajk = akiakj = <5,y (4.2) deta = l. (4.3)~~

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 192 P. G. APPLEBY, B. R. DUFFY AND R. W. OGDEN From (3.8)j the constraint (4.3) can also be expressed as

a a a e = e 4 4 hh hh • • • isis h...iN ii-is> ( - ) where £,-,.../„ are the Cartesian components of the volume element e given by (3.7). Examples of isotropic Cartesian tensors are the metric (or unit) tensor 1, with Cartesian components <5,y, and the tensor e itself. We write

~~'U'i'2 2 —iuiu~~

We now provide a new derivation of the form of the components Tilh_/(/ of an isotropic Cartesian tensor, considering separately the cases of even and odd U. (a) U is even. Because of the constraints on the components atj we cannot differentiate (4.1) as it stands along the lines used in Section 2 to obtain (2.4). We first need to incorporate the constraints in a suitable way with Lagrange multipliers. From (4.2) we form an isotropic tensor constraint of order U, namely

~~a a a hJ, hh • • • wiu 2 ^AO(/1;2...;U)= X K&o(ili2...iu)> (4.6) a e S(jS S~~

where the ACT are arbitrary scalars. If N is odd no product of a single alternating symbol with Kronecker deltas will yield an isotropic tensor of order U so that the constraint (4.4) is not required. We examine this case first. (i) N is odd. Equations (4.1) and (4.6) are combined to give the identity

~~= a a a ^inV.'o" ZJ ko&o(ili2...iu) iih i2j2- • • iuJu\Tjui...ju~ 2J ^o&o(ju2-..ju) | • oeSu <• oeSy J~~

~~With the ACT's regarded as Lagrange multipliers we can now differentiate this with respect to am,Ml, am2n2,..., amunu in turn to give~~

~~,im 6h . . . bium\ T - 2 AaAa( ) = 0, (4.7)~~

where the first summation is over all permutations rxr2 . . . rv = p(12 . . . U). If U < N then we may set /„ = ma = ar(l « a =£ U). The only non-vanishing contribu- tion to the first sum in (4.7) therefore arises when ra - a{\ =s a =£ U), and (4.7) then simplifies to

4 ' n1n2..nu~ Zj "CT^a(n,n2...ny)- V -") aeSu This result also holds when U > N, as can be shown by applying a similar argument to that used in the case of tensors of type (U, U) in Section 2. (ii) N is even. If U < N the most general form of isotropic Cartesian tensor again has components given by (4.8). If U = N the representation (4.8) is generalized to

*/i1n2...nw~ Z-i ^o&o(n\n2...nN)' \*'°n\n1...nN! v*'"/

* n\nz-..nu~ ZJ "•o&o(n\rt2...nu)' Zj MCT^o(ni)o(n!)...o(njv)^o(iijv+r)<'(«lv+2) ••"(«(;)' V^-*^) oeSu oeSy

where o(ni)o(n2) • • • o{nu) = a(n1n2 . . . nv) and the fia are Lagrange multipliers. Note that because of (3.11), specialized to the Cartesian case, at most one e term is required in each term in the second summation in (4.10). (b) U is odd. (i) N is even. In this case we may choose a,-,- = -6,y and (4.1) then yields T = (-1)UT • = -T 1 I]'2-'U V l' h'2-'U l Il'2-W' i.e. T is necessarily the zero tensor. (ii) N is odd. We form the product

"i\i2—iujih--JN~ iih—iuei\J2—JN V^--11/ to yield the components of an isotropic Cartesian tensor of even order U + N. By (a)(i) these components can be written in terms of products of \(U + N) Kronecker deltas. On multiplying (4.11) by ehjl_jN we obtain l Tllh...iu = (N\r Rilh..,uhh...jNehh...iN. (4.12)

~~If U~~

~~^ ii<2---'/v ~ ^e>ih—in> where A is an absolute scalar, thus specializing the corresponding result given in Section 3. If U> N then (4.12) yields~~

~~x e Tilil...iu= 2J l o o(il)o(i2)...o(iN)&o(iN+l)o(iN+2)...o(iu)> (4.13)~~

oeSN as in the second summation in (4.10). In conclusion we see that the most general form for the components of an isotropic Cartesian tensor is given by (4.10) with some or all of the coefficients being zero depending on whether U and N are even or odd. For another proof see [3] and the references given therein. It is not clear from [3] that all the separate cases discussed here have been covered previously. It is interesting to compare the results for Cartesian tensors with those for general tensors given in Section 2. In particular, for (7 = 4 and N odd we have, from (4.8),

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 194 P. G. APPLEBY, B. R. DUFFY AND R. W. OGDEN The 24 permutations of i, j, k, I yield three independent products of two Kronecker deltas, and we rewrite this equation as

~~4 14 T,,u = ad^u + 06*6,1 + Y&Ak> ( - )~~

where ex, /3, y are linear combinations of the 24 ACT's. This should be contrasted with (2.7), which holds for both odd and even N. It should be noted that in general not all products of the form Ahh...iMehh yjv (M even) are independent. This can be seen immediately by specializing (2.10) to the Cartesian case to give

~~2 (sgn o)&ilt,UiV2°(h>...Wr) = ° (r&N+l), aeSr~~

~~and then contracting with ehh,,,JN. For discussion of this in the context of elastic moduli (N = 3)see[6].~~

5. Isotropic tensor fields. Let M be a differentiate manifold of dimension /V2=2 and let TpM denote the tangent space at a point p eM. A tensor field of type (U, V) is a section of the tensor bundle ~~v TM = TM ® . . . ® TM ® TM* . . . TM*,~~

~~U times V times i.e. a map T:M-^®vTM which assigns to each point peM a tensor T(p) e~~

P Let us suppose that T is a general isotropic tensor field in the sense that, at each point p eM, T(p) is an isotropic tensor under GL(N). It follows immediately from Section 2 that U = V. Further, since the set of general isotropic tensors of type (U, U) forms an invariant vector space over M generated by the invariant isotropic tensors with components A}|;;;}{{, we can regard T as a vector-valued function over M. Hence the components of T can be written in the form (2.8) with the Ao now being scalar fields over M. We define the derivative of the isotropic tensor field T of type (U, U) to be the tensor field on M of type (U, U + 1), denoted by DT and having components!

~~1 0 1 i...juk ~ k i\-w £J ark^ji-)u y - )~~

~~relative to any local coordinate system (*'). By contrast, for any affine connection F on M, the covariant derivative DrT of T has~~

~~a . . t Note, however, that T-£ T'J\\\\V are not in general the components of a tensor field if T is not isotropic.~~

where rj;;;;J£|« denotes the expression on the left-hand side of (3.12) for W = 0 and V = U. Since T is isotropic the latter term on the right-hand side of (5.2) vanishes, and we obtain DrT = DT. (5.3) From (5.3) we deduce that homogeneity of general isotropic tensor fields is independent of any affine structure on M. Next we show that if v is any differentiate tangent vector field on M then the Lie derivative of T with respect to v can be written

~~LvT = (v.D)T (5.4)~~

~~when T is isotropic, where v.Dr(-) = t/[Dr(-)]* and similarly for v.D. To establish (5.4) we note (see [1], for example) that~~

~~L l V lTh-'u l \ 'v )l,...Ju fak )\-iu)\-iu fan i\-)~~

and, as in the case of (5.2), the latter term in (5.5) vanishes when T is isotropic. We now turn to tensor fields that are isotropic under SL{N). Suppose that M is now an orientable manifold endowed with a volume form e [1] (for example, M could be an incompressible continuous body); then the set of tensors that are isotropic under SL(N) forms an invariant vector space over M. Let :M—>M. be an isochoric (volume preserving) diffeomorphism. Then the tangent map T defines a canonical mapping of isotropic tensors at p e M (under SL(N)) onto the isotropic tensors at

~~DrT = DT+WT®Y, (5-6)~~

Y being given by Dre = -e <8> y when e is the particular volume form with components (3.7) and satisfying De = 0. This reduces to (5.3) whenever Dre = O, i.e. when F is 'volume preserving' (or compatible with the volume form). Further, for any tangent vector field v and volume form e,

~~LVT = v.DT - W(dive v)T, (5.7)~~

~~where, without the need for a connection, divev is defined by~~

~~Lve = (dive v)e.~~

Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 15:33:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006832 196 P. G. APPLEBY, B. R. DUFFY AND R. W. OGDEN The derivatives used above have an important role in continuum mechanics and can be used to simplify certain time derivatives. For example, if T is a material tensor field isotropic under GL(3) it follows from (5.3) and (5.4) that the material time derivative

~~is identical to the "convected" derivative 5T_5T 6t~ dt+UrT'~~

where v is the velocity vector field and Dr is the natural gradient operator in the reference frame. Hence the material time derivative of an isotropic material tensor field is an objective quantity. A similar conclusion holds when T is isotropic under SL(3), relative to the instantaneous volume form imposed on the body manifold by the reference frame, except that in this case the material time derivative and convected derivative satisfy the (objective) relation

~~6t where Dr.v = dive v + y.v.~~

REFERENCES 1. R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, tensor analysis and applications (Addison Wesley, 1983). 2. S. Golab, Tensor calculus (Elsevier, 1974). 3. H. Jeffreys, On isotropic tensors, Proc. Cambridge Philos. Soc. 73 (1973), 173-176. 4. M. S. Knebelman, Tensors with invariant components, Ann. of Math. (2) 30 (1928), 339-344. 5. J. G. Oldroyd and B. R. Duffy, Physical constants of a flowing continuum, /. Non- Newtonian Fluid Mech. 5 (1979), 141-145. 6. R. W. Ogden, On isotropic tensors and elastic moduli, Proc. Cambridge Philos. Soc. 75 (1974), 427-436. 7. J. A. Schouten, Tensor analysis for physicists, 2nd Edition (Oxford University Press, 1954). 8. T. Y. Thomas, Tensors whose components are absolute constants, Ann. of Math. (2) 27 (1926), 548-550.

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